We start the quantum annealing procedure with a quantum system that is driven by the transverse magnetic field \({\mathbf {H}}_X\)  and is initialized in its ground state \(|{\mathbf {v}}_+ \rangle \) .
The system then gradually evolves from the initial Hamiltonian \({\mathbf {H}}_X\)  to its final target Hamiltonian \({\mathbf {H}}_I^q\) .
During the Hamiltonian change, the system tends to stay in the ground states of the instantaneous Hamiltonian via quantum tunneling based on the adiabatic quantum theorem [1]}, [2]}, [3]}, [4]}.
At the end of the annealing procedure, if the quantum system stays in a ground state of the final Hamiltonian \({\mathbf {H}}_I^q\) , we are able to obtain an optimal solution by measuring the system.
